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Theorem elrnmpt2d 5804
 Description: Elementhood in the range of a function in maps-to notation, deduction form. (Contributed by Rohan Ridenour, 3-Aug-2023.)
Hypotheses
Ref Expression
elrnmpt2d.1 𝐹 = (𝑥𝐴𝐵)
elrnmpt2d.2 (𝜑𝐶 ∈ ran 𝐹)
Assertion
Ref Expression
elrnmpt2d (𝜑 → ∃𝑥𝐴 𝐶 = 𝐵)
Distinct variable group:   𝑥,𝐶
Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)   𝐵(𝑥)   𝐹(𝑥)

Proof of Theorem elrnmpt2d
StepHypRef Expression
1 elrnmpt2d.2 . 2 (𝜑𝐶 ∈ ran 𝐹)
2 elrnmpt2d.1 . . . 4 𝐹 = (𝑥𝐴𝐵)
32elrnmpt 5797 . . 3 (𝐶 ∈ ran 𝐹 → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥𝐴 𝐶 = 𝐵))
43ibi 270 . 2 (𝐶 ∈ ran 𝐹 → ∃𝑥𝐴 𝐶 = 𝐵)
51, 4syl 17 1 (𝜑 → ∃𝑥𝐴 𝐶 = 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  ∃wrex 3071   ↦ cmpt 5112  ran crn 5525 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-rex 3076  df-v 3411  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-sn 4523  df-pr 4525  df-op 4529  df-br 5033  df-opab 5095  df-mpt 5113  df-cnv 5532  df-dm 5534  df-rn 5535 This theorem is referenced by:  ablsimpg1gend  19295
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